Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems

Abstract

Some discontinuous Galerkin methods for the linear convection-diffusion equation −ε u″+bu′=f are studied. Based on superconvergence properties of numerical fluxes at element nodes established in some earlier works, e.g., Celiker and Cockburn in Math. Comput. 76(257), 67–96, 2007, we identify superconvergence points for the approximations of u or q=u′. Our results are twofold:

1) For the minimal dissipation LDG method (we call it md-LDG in this paper) using polynomials of degree p, we prove that the leading terms of the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p+1, respectively. Consequently, the zeros of the right-Radau and left-Radau polynomials of degree p+1 are the superconvergence points of order p+2 for the discretization errors of the potential and of the gradient, respectively.

2) For the consistent DG methods whose numerical fluxes at the mesh nodes converge at the rate of O(h ^p+1), we prove that the leading term of the discretization error for q is proportional to the Legendre polynomial of degree p. Consequently, the approximation of the gradient superconverges at the zeros of the Legendre polynomial of degree p at the rate of O(h ^p+1).

Numerical experiments are presented to illustrate the theoretical findings.